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August  2018, 38(8): 3851-3873. doi: 10.3934/dcds.2018167

## Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle

 Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, 525-8577, Japan

*Corresponding author

1Present address: Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-22100 Lund, Sweden

Received  September 2017 Revised  February 2018 Published  May 2018

Bohr-Sommerfeld type quantization conditions of semiclassical eigenvalues for the non-selfadjoint Zakharov-Shabat operator on the unit circle are derived using an exact WKB method. The conditions are given in terms of the action associated with the unit circle or the action associated with turning points following the absence or presence of real turning points.

Citation: Setsuro Fujiié, Jens Wittsten. Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3851-3873. doi: 10.3934/dcds.2018167
##### References:
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##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces , vol. 140 of Pure and Applied Mathematics, Elsevier/Academic Press, 2003.  Google Scholar [2] N. Dencker, The pseudospectrum of systems of semiclassical operators, Analysis & PDE, 1 (2008), 323-373.  doi: 10.2140/apde.2008.1.323.  Google Scholar [3] S. Dyatlov and M. Zworski, Stochastic stability of Pollicott-Ruelle resonances, Nonlinearity, 28 (2015), 3511-3533.  doi: 10.1088/0951-7715/28/10/3511.  Google Scholar [4] J. Ecalle, Cinq Applications des Fonctions Résurgentes , Prépublications mathématiques d'Orsay, Département de mathématique, 1984. Google Scholar [5] M.A. Evgrafov and M.V. Fedoryuk, {Asymptotic behaviour as $λ\to∞$ of the solution of the equation $w''(z)-p (z, λ) w (z) = 0$ in the complex $z$-plane, Russian Mathematical Surveys, 21 (1966), 3-50.   Google Scholar [6] S. Fujiié, C. Lasser and L. Nédélec, Semiclassical resonances for a two-level Schrödinger operator with a conical intersection, Asymptotic Analysis, 65 (2009), 17-58.   Google Scholar [7] S. Fujiié and T. Ramond, Matrice de scattering et résonances associées à une orbite hétérocline, Ann. Inst. H. Poincaré Phys. Théor., 69 (1998), 31-82.   Google Scholar [8] S.V. Galtsev and A.I. Shafarevich, Quantized Riemann surfaces and semiclassical spectral series for a non-self-adjoint Schrödinger operator with periodic coefficients, Theoretical and Mathematical Physics, 148 (2006), 1049-1066.  doi: 10.1007/s11232-006-0100-y.  Google Scholar [9] C. Gérard and A. Grigis, Precise estimates of tunneling and eigenvalues near a potential barrier, Journal of Differential Equations, 72 (1988), 149-177.  doi: 10.1016/0022-0396(88)90153-2.  Google Scholar [10] B. Grébert and T. Kappeler, Estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system, Asymptotic Analysis, 25 (2001), 201-237.   Google Scholar [11] K. Hirota, Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov-Shabat operator , Journal of Mathematical Physics, 58 (2017), 102108, 14pp. doi: 10.1063/1.4999668.  Google Scholar [12] T. Kato, Perturbation Theory for Linear Operators, vol. 132 of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer Verlag, 1966.  Google Scholar [13] S. Kmvissias, K. McLaughlin and P. Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation (AM-154), 154, Princeton University Press, 2003. doi: 10.1515/9781400837182.  Google Scholar [14] E.L. Korotyaev and P. Kargaev, Estimates for periodic Zakharov-Shabat operators, Journal of Differential Equations, 249 (2010), 76-93.  doi: 10.1016/j.jde.2010.02.016.  Google Scholar [15] T. Ramond, Semiclassical study of quantum scattering on the line, Communications in Mathematical Physics, 177 (1996), 221-254.  doi: 10.1007/BF02102437.  Google Scholar [16] A.B. Shabat and V.F. Zakharov, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Journal of Experimental and Theoretical Physics, 34 (1972), 62-69.   Google Scholar [17] M. Zworski, Semiclassical Analysis, vol. 138 of Graduate Studies in Mathematics, American Mathematical Society, 2012. doi: 10. 1090/gsm/138.  Google Scholar
The configuration of Stokes lines for V (x) = cos x, with legends describing the size of Re z(x; 0, λ) for λ = 1 (left panel) and λ = 1 + 10−1i (right panel)
The configuration of Stokes lines and the behavior of Re z(x; x1; λ) for V (x) = cos x, where λ = and x1 satisfies Re x1 > 0, cos x1 = µ (the turning point in the middle). The top panel describes the situation for µ = 1/2 and the bottom panel for µ = 1/2 + i/10. Branch cuts are located along (the curved edges of) the white regions.
The location of amplitude base points relative the neighboring turning points $x_k(\mu)$ over a partial period for generic $V$ and $\lambda = i\mu\in B_\varepsilon(\lambda_0)$. Branch cuts are indicated by dashed lines.
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